So you, as the ancient philosopher in mathematics
have concluded in order for the multiplication of positive and negative numbers to be
consistent with everything you've been constructing so far
with all the other properties of multiplication that you know so far
that you need a negative number times a positive
number or a positive times a negative to give you a negative number
and a negative times a negative
to give you a positive number and so you accept
it's all consistent so far.. this deal does not make complete concrete
sense to you, you want to have a slightly deeper institution than just having to accept its
consistent with the distributive property and whatever else and so you try another
thought experiment, you say "well what is just a basic multiplication way of doing it?"
So if I say, two times
three, one way to
to conceptualize is basic multiplication is really repeating
addition, so you could view this as two threes
so let me write three plus three
and notice there are two of them, there are two of these
or you could view this as three twos, and so this is the same thing as
two plus two plus two and there are
three of them, and either way you can conceptualize
as you get the same exact answer. This is going to be equal
to six, fair enough!
Now, you knew this before you even tried to tackle negative numbers.
Now let's try to make one of these negatives and see what
happens. Let's do two
times negative
three, I want to make the negative into a different color. Two times
negative three.
Well, one way you could view this is the same analogy
here, it's negative three twice so it would be
negative.. I'll try to color code it
negative three and then another negative
three or you could say negative three minus three
or, and this is the interesting thing, instead of
over here there's a two times positive three
you added two, three times.
But since here is two times negative three
you could also imagine you are going to subtract two, three times
So instead of up here, I could
written two plus two plus two because this is a positive
two right over here, but since we're doing this over negative three
we could imagine subtracting two, three times, so this would be
subtracting two (repeated)
subtract another two right over here, subtract another two
and then you subtract another two
notice you did it, once again, you did it
three times, so this is a negative three, so essentially you are subtracting
two, three times. And either way, you can conceptualize
right over here, you are going to get negative six
negative six is the answer.
Now, so you are already starting to feel better about this part right over here
negative times a positive, or a positive times a negative
is going to give you a negative. Now lets take to the really un-intuitive one
and measure negative times a negative, and all of a sudden negatives kind of cancel
to give you a positive. Now why is that the case? Well we can just build from
this example right over here. Let's say we had
a negative two, lets say we had
negative two, let me do it a different color,
let's say we had a negative two, I already used this color
negative two times
negative three.
So now, we can d- actually I'll do this one first.
Let's do multiplying something by negative three so we'll
repeatedly subtract that thing three times whatever that thing is
so now the thing isn't a positive two so the thing over
here is a positive two but the thing we're going to subtract is a negative two
So let me make it clear, this says we are going to subtract something
three times, so we subtract something three times, so
subtracting something (repeatedly) three times
That's what this part right over here tells us
and we'll do this, exactly three times
Over here, it was a positive two we subtracted three times, now we're going to
do a negative two, now we're going to do a negative two
and we know from subtracting negative numbers, we already
built this intuition that subtracting a negative is the same thing
it's the same thing as adding a positive, and so this
this is going to be the same thing as two plus two plus two and
we're told once again, gives you a positive
six, you can same use the same logic over here, now
instead of adding negative three twice, really I could have written this as
negative three as this example
negative three
negative three, and we added it
we added it, now let me put a plus here to make it clear
over here we added it twice, we added negative three
two times, or here since we have a negative two, we're going to subtract
to negative three twice, so we're going to subtract something
and we're going to subtract something again, and that something is going to be
our negative three, it's going to be our negative three, so
negative, negative and put our three right over here
and once again, subtracting negative three is like taking away
someone's debt, which is essentially giving them money,
this is the same thing as adding three plus three which is once again six. So now
you, the ancient philosopher, feel pretty good. Not only this
all consistent with all the mathematics you know
the distributive property is also the property of multiplying something
times something all these things you already know, and now
this actually makes conceptual sense to you, this is actually very consistent with
with your notations, your original notations, or one of the positive notations
of multiplication which is as repeated addition